A176232 Determinant of the n X n matrix with rows (1!,-1,0,...,0), (1, 2!,-1,0,...,0), (0,1,3!,-1,0,...,0), ..., (0,0,...,1,n!).

1, 1, 3, 19, 459, 55099, 39671739, 199945619659, 8061807424322619, 2925468678338137602379, 10615940739961495538937237819, 423754383328897950597328272711061579, 202979027621555455188781938315330372976764219

Offset: 0

Author

Michel Lagneau, Apr 12 2010

Keywords

• nonn

Comments

Each determinant is the numerator of the fraction x(n)/y(n) = [1!, 2!, ..., n!] (the simple continued fraction). The value x(n) is obtained by computing the determinant det(n X n) from the last column. The value y(n) is obtained by computing this determinant after removal of the first row and the first column (see example below).

Also denominator of fraction equal to the continued fraction [ 0; 1!, 2!, ... , n! ]. - Seiichi Manyama, Jun 05 2018

Examples

			For n = 1, det[1] = 1.
For n = 2, det([[1,-1],[1,2]]) = 3, and the continued fraction expansion is 3/2 = [1!,2!].
For n = 3, det([[1,-1, 0],[1,2,-1],[0,1,6]]) = 19, and the continued fraction expansion is 19/det([[2,-1],[1,6]]) = 19/13 = [1!,2!,3!].
For n = 4, det([[1,-1,0,0],[1,2,-1,0],[0,1,6,-1],[0,0,1,24]]) = 459, and the continued fraction expansion is 459/det([[2,-1,0],[1,6,-1],[0,1,24]]) = 459/314 = [1!,2!,3!,4!].
		

References

• J. M. De Koninck, A. Mercier, 1001 problèmes en théorie classique des nombres. Collection ellipses (2004), p.115.

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..43

Crossrefs

Cf. A001040, A036245, A036246, A084845.

Programs

Maple

for n from 15 by -1 to 1 do:x0:=n!:for p from n by -1 to 2 do : x0:= (p-1)! + 1/x0 :od:print(x0):od :

Formula

a(0) = 1, a(1) = 1, a(n) = n! * a(n-1) + a(n-2). - Daniel Suteu, Dec 20 2016

a(n) ~ c * BarnesG(n+2), where c = 1.5943186620010986362991550255196986158205795892595646967623357407966... - Vaclav Kotesovec, Jun 05 2018

Extensions

a(0)=1 prepended by Alois P. Heinz, Dec 20 2016